Image Processing Reference
In-Depth Information
v
t
If the curve is parameterised by arc length, then
|()|
is constant. Thus, the derivative of
a tangential vector is simply given by
˙˙
v
v
˙
() = | ()|
t
t
(- sin (
ϕ
(
t
)) +
j
cos (
ϕ
(
t
))) (
d
ϕ
(
t
)/
dt
)
(4.36)
Since we are using a normalised parameterisation, then
d
ϕ
(
t
)/
dt
=
d
ϕ
(
t
)/
ds
. Thus, the
tangential vector can be written as
˙˙
v
() =
t
κ
()
t
n
()
t
(4.37)
v
t
(
t
))) defines the direction of
˙
v
(
t
whilst the curvature
where
n
(
t
) =
|()|
(-sin(
ϕ
(
t
)) +
j
cos(
ϕ
κ
(
t
) defines its modulus. The derivative of the normal vector is given by
n
v
˙
( ) = |
t
( ) | (- cos(
t
ϕ
( )) - sin(
t
j
ϕ
(
t
)))(
d
ϕ
( )/
t
ds
)
that can be written as
n
˙
κ
v
(4.38)
Clearly
n
(
t
) is normal to
v
(
t
. Therefore, for each point in the curve, there is a pair of
orthogonal vectors
v
(
t
and
n
(
t
) whose moduli are proportionally related by the curvature.
Generally, the curvature of a parametric curve is computed by evaluating Equation 4.34.
For a straight
line
, for example, the second derivatives
˙˙
x
() and
˙˙
y
() are
zero
, so the
curvature function is
nil
. For a
circle
of radius
r
, we have
˙
x
() =
r
cos(
t
) and
˙
y
() = -
r
sin(
t
). Thus,
˙˙
y
() = -
r
cos(
t
),
˙˙
x
() = -
r
sin(
t
) and κ (
t
) = 1/
r
. However, for curves in digital
images, the derivatives must be computed from discrete data. This can be done in four
main ways. The most obvious approach is to calculate curvature by directly computing the
difference between angular direction of successive edge pixels in a curve. A second approach
is to apply the curvature function to a continuous approximation to the discrete data. In a
third approach, a measure of curvature can be derived from changes in image intensity.
Finally, a measure of curvature can be obtained by correlation.
() = - () ()
t
t
t
4.6.1
Computing differences in edge direction
Perhaps the easier way to compute curvature in digital images is to measure the
angular
change
along the curve's path. This approach was considered in early corner detection
techniques (Bennett, 1975), (Groan, 1978), (Kitchen, 1982) and it merely computes the
difference
in edge
direction
between connected pixels forming a discrete curve. That is, it
approximates the derivative in Equation 4.30 as the difference between neighbouring pixels.
As such, curvature is simply given by
k
(
t
) = ϕ
t
+1
- ϕ
t
-1
(4.39)
where the sequence . . . ϕ
t
-1
, ϕ
t
, ϕ
t
+1
, ϕ
t+2
. . . represents the gradient direction of a sequence
of pixels defining a curve segment. Gradient direction can be obtained as the angle given
by an edge detector operator. Alternatively, it can be computed by considering the position
of pixels in the sequence. That is, by defining ϕ
t
= (
y
t
-1
-
y
t
+1
)/(
x
t
-1
-
x
t
+1
) where (
x
t
,
y
t
)
denotes pixel
t
in the sequence. Since edge points are only defined at discrete points, this
angle can only take eight values, so the computed curvature is very ragged. This can be
smoothed out by considering the difference in mean angular direction of
n
pixels on the
leading and trailing curve segment. That is,
n
-1
() =
1
-
1
kt
ϕ
ϕ
(4.40)
n
n
n
ti
+
n
ti
+
i
=1
i
=-
